MinT - Best Known (162−35, 162, s)-Nets in Base 4

Best Known (162−35, 162, s)-Nets in Base 4

(162−35, 162, 1048)-Net over F₄ — Constructive and digital

Digital (127, 162, 1048)-net over F₄, using

4² times duplication [i] based on digital (125, 160, 1048)-net over F₄, using
- trace code for nets [i] based on digital (5, 40, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(162−35, 162, 3772)-Net over F₄ — Digital

Digital (127, 162, 3772)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶², 3772, F₄, 35) (dual of [3772, 3610, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁶², 4119, F₄, 35) (dual of [4119, 3957, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(30) [i] based on
    1. linear OA(4¹⁵⁷, 4096, F₄, 35) (dual of [4096, 3939, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(4¹³⁹, 4096, F₄, 31) (dual of [4096, 3957, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(4⁵, 23, F₄, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(162−35, 162, 1204151)-Net in Base 4 — Upper bound on s

There is no (127, 162, 1204152)-net in base 4, because

1 times m-reduction [i] would yield (127, 161, 1204152)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 543953 995491 133289 182847 566574 779835 927418 252680 825613 463376 058355 977849 884135 062835 957984 542955 > 4¹⁶¹ [i]